I want to evaluate
$$\int_0^{\infty}\frac{e^{jtx}}{(1+x)^2}\,dx$$
where $j=\sqrt{-1}$ and $t$ is a real number. I did this change of variables $y=1/(1+x)$, which resulted in the integration
$$-\int_0^1e^{jt/y}\,dy=\frac{1}{jt}$$
As you can see, the original integral is actually the characteristic function of a random variable with a PDF $f_X(x)=(1+x)^{-2}$, and one of the properties of the characteristic function is that its value at $0$ is $1$, which isn't the case in my result, and the characteristic function always exists.
What's wrong with the derivation?
EDIT 1: I evaluated $dy=\frac{dx}{(1+x)^2}$, while it should be $dy=\frac{-dx}{(1+x)^2}$. So, the integral becomes
$$\int_0^1e^{jt/y}\,dy=\frac{-1}{jt}$$
EDIT 2: I forgot to the scaling factor $e^{jt}$, but the final result is still true. So, the integral is
$$e^{jt}\int_0^1e^{jt/y}\,dy=\frac{-1}{jt}$$
EDIT 3: The indefinte integral is evaluated as
$$e^{jt}\int e^{jt/y}\,dy = \frac{-e^{jt}}{jt}\,y^2e^{jt/y}$$
You evaluate the integral as if the function was $e^{jty}$.